This paper concerns projectively Anosov flows ${\phi }^t$ with smooth stable and unstable foliations ${ℱ}^s$ and ${ℱ}^u$ on a Seifert manifold $M$. We show that if the foliation ${ℱ}^s$ or ${ℱ}^u$ contains a compact leaf, then the flow ${\phi }^t$ is decomposed into a finite union of models which are defined on $T^2\times I$ and bounded by compact leaves, and therefore the manifold $M$ is homeomorphic to the 3-torus. In the proof, we also obtain a theorem which classifies codimension one foliations on Seifert manifolds with compact leaves which are incompressible...

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1,2,\cdots ,\infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibitinglargerobustly transitive sets. We show that the $C^{\infty }$ closure of such open sets contains a variety of systems, including so-calleda priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of...

Nous présentons une méthode permettant d’établir le théorème limite central avec vitesse en $n^{-1/2}$ pour certains systèmes dynamiques. Elle est basée sur une propriété de décorrélation forte qui semble assez naturelle dans le cadre des systèmes quasi-hyperboliques. Nous prouvons que cette propriété est satisfaite par les exemples des flots diagonaux sur un quotient compact de $\mathrm{SL}(d,ℝ)$ et les « transformations » non uniformément hyperboliques du tore ${𝕋}^3$ étudiées par Shub et Wilkinson.